### Quantifiers and logical inference

```Adapted from Patrick J. Hurley, A
Concise Introduction to Logic (Belmont:
Before I go on to explain quantifiers, first let me
address different ways of symbolizing statements.
Previously, we used one letter to symbolize one
statement. But there is another way to symbolize
certain kinds of statements that are relevant to
quantifiers. We can also symbolize statements by
symbolizing the predicate and subject separately.
E.g. Luna is a cat. = Cl (C= is a cat; l=Luna)
The relationship has a parallel in functional or
quantificational statements. Luna is only an
instance of a cat. There can be many cats. ‘Cats’ is
the universal category governing whether a
constant or variable counts as one of the things
the term refers to, and Luna is the particular
instantiation or constant.
Instead of ‘Luna is a cat’, what if I wanted to say
something about all cats or some cats as a
category? This is when we use quantification.
Let’s say I want to say ‘All cats are animals’. The
symbol Ac does not cut it because the lower case
‘c’ represents only an individual. When I want to
say something about all cats (or no cats), I need to
use the symbol (x).
x symbolizes the quantifier in universal
statements:
All cats are animals. =
(x)(Cx  Ax)
(Translation: For every x, if x is a cat, then x is an
animal.)
No cats are dogs.
=
(x)(Cx   Dx)
(Translation: For every x, if x is a cat, then x is not
a dog.)
The previous slide contains statements about x
such that for EVERY x, if x is a cat, then it is an
animal. It is not just about one individual, but
about the whole category of cats, dogs and
animals.
See the next slide for Existential Statements.
x symbolizes the quantifier in existential
statements.
Some apples are red. = (x)(Ax  Rx)
(Translation: There exists an x such that x is an
apple and x is red.)
Some apples are not red. = (x)(Ax  Rx)
(Translation: There exists an x such that x is an
apple and x is not red.)
Statement









There are happy marriages.
Every pediatrician loses sleep.
Animals exist.
Unicorns do not exist.
Anything is conceivable
Sea lions are mammals.
Egomaniacs are not pleasant
companions.
A few egomaniacs did not arrive
on time.
Only close friends were invited
to the wedding.
Symbolic translation
(x) (Mx • Hx)
(x) (Px ⊃ Lx)
(x)Ax
~(x)Ux
(x)Cx
(x) (Sx ⊃ Mx)
(x) (Ex ⊃ ~Px)
(x) (Ex • ~Ax)
(x) (Ix ⊃ Cx)








Translate the following statements into symbolic
form. Avoid negation signs preceding quantifiers.
The predicate letters are given in parentheses.
1. Elaine is a chemist. (C)
2. Nancy is not a sales clerk. (S)
3. Intel designs a faster chip only if Micron does.
(D)
4. Some grapes are sour. (G, S)
5. Every penguin loves ice. (P, L)
6. There is trouble in River City. (T, R)
7. Tigers exist. (T)
There are a number of logical inferences that we
use in every day language, logic, and math. Logic
has names for them. Once you identify them, it
makes those inferences explicit. Your textbook
mentions six such inferences, and I will give them
labels: modus ponens, modus tollens, disjunctive
syllogism, conjunction, simplification, and
We have run across this inference before. The rule
of modus ponens is that if you have a conditional
and assert the antecedent, you may infer the
consequent.
E.g.
If it rains, then the ground gets wet.
It is raining.
Therefore, the ground is wet.
This inference can be symbolized this way:
RW
R
W
This is an inference that also works off a
conditional, but it asserts something about the
consequent. If you have a conditional, and assert
the negation of the consequent, you can infer the
negation of the antecedent.
E.g.
If it rains, then the ground gets wet.
The ground did not get wet.
Therefore, it did not rain.
This inference can be symbolized this way:
RW
 W
 R
In the disjunctive syllogism, if you have a
disjunction and negate one side of the disjunctive,
you may infer the non-negated side of the
disjunctive.
E.g.
Either the chalk is black or the chalk is white.
The chalk is not black.
Therefore, the chalk is white.
This inference can be symbolized this way:
BVW
~B
W
In this type of inference, if you have two
established statements, then you may also conjoin
them to make a single statement.
E.g.
Chalk is white. Snow is white. Chalk and snow
are white.
This inference may be symbolized this way:
C
S
CS
In this inference, if you have two statements that
are conjoined into a complex statement, you may
infer one part of the complex statement.
E.g.
Chalk and snow are white.
Chalk is white.
This inference may be symbolized this way:
CS
C
In this inference, if you have a statement, you
may properly add a disjunction without falsifying
the original statement.
E.g.
Chalk is white.
Chalk is white or Rudolf ate my homework.
The previous inference can be symbolized this
way:
C
CVR
Fill in the missing premise and give the inference that
justifies the conclusion.
 (1) 1. B v K
2. ______
3. K ____


(2) 1. N  S
2. ______
3. S ____
(3) 1. K  T
2. ______
3. ~K ____



(1) 1. ~A
2. A v E
3. ______ ____
4. ~A  E ____
(2) 1. T
2. T  G
3. (T v U)  H
4. ___________ ____
5. H ____
(3) 1. M
2. (M  E)  D
3. E
4. __________ ____
5. D ____
```